Reflected Brownian motion in Weyl chambers

TL;DR

This paper provides two descriptions of the pushing process in reflected Brownian motion within Weyl chambers, revealing new insights into its structure and connections to particle systems and skew Brownian motion.

Contribution

It introduces novel descriptions of the pushing process in Weyl chamber reflected Brownian motion, including a sum of local times and a multivoque SDE formulation.

Findings

Characterization of the pushing process via local times and orbits.

Representation of the process as a sum of inward normal vectors.

Construction of a multidimensional skew Brownian motion.

Abstract

We supply two different descriptions of the pushing process driving the reflected Brownian motion in Weyl chambers, when the latter domains are simplexes. The first one shows that a simple root lies in one and only one orbit if and only if the pushing process in the direction of that simple root increases as the sum of all the Brownian local times in the directions of the orbit's positive elements. The last one shows that the pushing process may be written as the sum of an inward normal vector at the chamber's boundary and an inward normal vector at the origin, yielding a kind of a multivoque stochastic differential equation for the reflected process. We finally give a particles system interpretation of the reflected process and construct a multidimensional skew Brownian motion.